# Exploring Amputated Connected Graphs in QFT

• PRB147
In summary: Srednicki simply defines the generating functional by a path integral (not even mentioning Feynman diagrams) and then in chapter 10 he says it is useful to redraw the diagrams in terms of 1PI instead of 2PI (which I think is related to amputation).In summary, amputated connected graphs in Peskin and Schroeder's book refer to graphs that have been truncated by removing external lines and keeping the momenta that flowed through them. This is related to the concept of one-particle irreducible (1PI) graphs, which are defined as diagrams that cannot be disconnected by cutting a single line. The term "irreducible" can also refer to proper diagrams, which cannot be separated into two disconnected
PRB147
In Peskin and Schroder Book QFT:
They used the term: "amputated connected graphs".
Does amputated connected graph in Peskins's book
is same as the irreducible graph in common sense?

I'm not sure what you mean by "irreducible graph in common sense".

In QFT you often talk about one-particle-irreducible graphs which are closely related to the quantum effective action. These are graphs that can not be disconnected by simply cutting a single line (propagator). Connected graphs (eg tree graphs) can be disconnected by cutting a single line. Amputated means that you remove the external lines but keep the momenta that flowed through them -- it is not related to connectedness.

Quoting from another text,

Consider any diagram M. A part of M which is connected to the rest of M by exactly two lines is called an inserted SE-part. Similarly, a part of M which is connected to the rest of M by exactly two electron lines and one photon line is called an inserted V-part. To every M there corresponds a uniquely defined diagram M' called the skeleton of M, which is obtained from M by replacing every inserted SE-part by a line, and every inserted V-part by a corner...

A diagram or diagram part is called irreducible if it is equal to its own skeleton. It is called reducible is this is not the case. A reducible SE- or V-part can be either proper or improper. It is called improper if it can be separated into two disconnected parts by the omission of one single line. Diagrams for which this is not the case are called proper.

Too many words with conflicting definitions. Too many physicists making up terminology.

Dy the way, which text is that? I don't think I've ever heard that terminology before.

I remember skeleton of a graph to mean the tree graph obtained obtained from a general graph by contracting every one-particle irreducible (1PI) subgraph to a single point.
The reverse of this (inserting 1PI graphs into a tree) is how you use 1PI graphs obtained from the effective action to construct arbitrary amplitude. This shows how the effective action contains all of the information that is in the full generating functional.

Thank you all for the replies.
my "irreducible" here means the proper.
see Peskin's Book, amputated graphs do have external lines, see the graph
in page 114 of their book.

@PRB147:

So irreducible means 1-particle irreducible. (Or in graph theory speak: 2-connected).

As for amputated graphs and external lines, it depends what you mean be external lines.
The diagram in P&S shows that you truncate all parts of the graph that make up contributions to external full propagators. This includes the "inner-most external line" which has to be a perturbative propagator. All that's left is the momentum that would have been running through the external propagators and the vertex that you're interested in. This external momenta does not really correspond a full external edge, since an edge --> propagator.

I've seen more mathematical texts talk about how the Feynman diagram way of creating graphs is useful. You start off with half-edges (fields) which are then joined to form full edges (Wick contract two fields to get a propagator). In this sense, the external lines are half-edges -- ie places where you can attach another half-edge in order to get the expression needed for scattering amplitudes.

This point of view works even better when thinking of the quantum effective action. Then the external lines correspond to the classical/external/background field - i.e. the arguments of the effective action.

Too many words with conflicting definitions. Too many physicists making up terminology. Dy the way, which text is that? I don't think I've ever heard that terminology before.
Now you have. I believe the physicist in question is Freeman Dyson.
I remember skeleton of a graph to mean the tree graph obtained obtained from a general graph by contracting every one-particle irreducible (1PI) subgraph to a single point
You got it partly right, but in general the skeleton is not a tree. Which is what gives rise to overlapping divergences.

I have a somewhat related question. If anyone is familiar with Srednicki's textbook (which takes the path integral approach), I am wondering where he accounts for need to amputate the diagrams, or what corresponding step is made. In Peskin and Schroeder the amputation is related to the LSZ formula, but Srednicki never mentions amputation in relation to the LSZ formula or Feynman diagrams (sections 5, 9-10 of Srednicki).

## 1. What is a connected graph in QFT?

A connected graph in QFT is a mathematical representation of a physical system that is made up of interconnected points or nodes. In QFT, these points represent the particles or fields in a physical system, and the connections between them represent the interactions between these particles or fields.

## 2. How are amputated connected graphs used in QFT?

Amputated connected graphs are used in QFT to calculate the scattering amplitudes of particles or fields in a physical system. These graphs represent the possible ways in which particles or fields can interact with each other, and by calculating their amplitudes, we can determine the likelihood of different interactions occurring.

## 3. What is the importance of exploring amputated connected graphs in QFT?

Exploring amputated connected graphs in QFT allows us to understand the underlying principles and mechanisms of particle interactions in a physical system. By studying these graphs, we can make predictions about the behavior of particles and fields and test them through experiments.

## 4. How are amputated connected graphs related to Feynman diagrams?

Feynman diagrams are a type of amputated connected graph that is used in QFT to visualize and calculate particle interactions. They were developed by physicist Richard Feynman and are based on the same principles as amputated connected graphs.

## 5. Are there any limitations to using amputated connected graphs in QFT?

While amputated connected graphs are a powerful tool in QFT, they do have some limitations. They can become very complex and difficult to calculate for systems with a large number of particles or fields. Additionally, they do not take into account certain quantum effects, such as virtual particles, that can play a role in particle interactions.

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